The emission of Be, C, O, and Ne clusters from seven parent nuclei with neutron numbers around the neutron magicities N = 82, and 126 are considered. The universal decay law (UDL) formula, as well as the double-folding model derived from M3Y-Paris NN interaction with zero- and finite-range exchange components, are utilized to compute the half-life time for 23 cluster decay processes. The calculations utilizing the UDL formula show satisfactory agreement with the experimental data. The reliable UDL formula is used to calculate log Tcfor more than 1500 cluster emitters and its variation with the neutron number, Nd, of the daughter nuclei is presented. The behavior of log Tc with neutron number variation is studied and correlated to the energy levels of the daughter nuclei. For neutron number Nd larger than the neutron magic number, log Tc increases almost linearly with increasing Ndleaving the daughter nuclei in most cases with the same nuclear spin value. This linear behavior of log Tc results from equal nuclear spin values of the daughter nuclei. At the magic neutron number, the nuclear spin changes strongly and as a result the behavior of log Tc is reversed, increases as Nd decreases and reaches to a maximum value at Nd value corresponds to emitting all the neutrons in the cluster from levels below the neutron magic number gap. Leaving the daughter nuclei in the same spin produces almost linear variation of log Tc. For protons in various clusters emitted from the same level or the same group of levels, log Tc has almost the same value and the same behavior of variation with Nd. Also, the values of log Tc for specific type of cluster depend on the N to Z ratio for different isotopes of this cluster. From the available nuclear spin values, the neutron energy levels around the magic numbers are presented.